The Labor Market Effects of Reducing the Number of Illegal Immigrants

Working Paper 04-2015

The Labor Market Effects of Reducing the Number of
Illegal Immigrants

Andri Chassamboulli and Giovanni Peri

  Department of Economics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
  Tel.: +357-22893700, Fax: +357-22895028, Web site:
The Labor Market Effects of Reducing the Number of
               Illegal Immigrants
                   Andri Chassamboulli (University of Cyprus)
                  Giovanni Peri (University of California, Davis)∗
                                      March, 23rd, 2015

          A controversial issue in the US is how to reduce the number of illegal immigrants
      and what effect this would have on the US economy. To answer this question we set
      up a two-country model with search in labor markets and featuring legal and illegal
      immigrants among the low skilled. We calibrate it to the US and Mexican economies
      during the period 2000-2010. As immigrants, especially illegal ones, have a worse
      outside option than natives their wages are lower. Hence their presence reduces the
      labor cost of employers who, as a consequence, create more jobs per unemployed
      when there are more immigrants. Because of such effect our model shows that
      increasing deportation rates and tightening border control weakens the low-skilled
      labor markets, increasing unemployment of native low skilled. Legalization, instead
      decreases the unemployment rate of low-skilled natives and it increases income per

         JEL codes: F22, J61, J64.
         Key Words: job creation, search costs, illegal immigrants, border controls,
      deportations, legalization, unemployment, wages.

    Andri Chassamboulli, Departement of Economics, University of Cyprus, CY-1678 Nicosia; an- Giovanni Peri, Department of Economics, UC Davis. One Shields Avenue, Davis Ca
95616 USA; We are grateful to three anonymous referees for their helpful comments.

1         Introduction
Most of the existing papers on the labor market effects of immigration consider the num-
ber and the skill composition of immigrants as an exogenous variable and analyze the
consequences of changing those on native labor market outcomes. The number and type
of immigrants entering a country, however, are not policy variables of choice, but the
outcomes of economic, social and policy forces in the sending and receiving countries. In
the economic literature on the effect of immigration, very little attention has been paid
to the specific policies used and to the difference between the labor market effects of legal
and illegal immigrants.1
        Large part of the policy debate in the US, however, has been about different ways
to reduce the number of illegal immigrants. The presence of a large number of illegal
immigrants is an anomaly, but there is disagreement on how to address it. A question often
asked to economists is whether reducing illegal immigrants would be costly or beneficial
for the US economy. In particular, what policy, among border enforcement, deportation,
self-deportation or legalization, would be most harmful to US firms and workers? The
existing economic analysis uses naive frameworks to answer this question. Based on
an oversimplified canonical model of labor demand and supply, economists rarely focus
explicitly on illegal immigrants and they overlook the different implications across policies.
The goal of this paper is to fill this gap by using a more insightful model to analyze
the effects of different policies aimed at reducing the number of illegal immigrants. Do
fewer illegal immigrants free jobs for Americans or do they reduce firm’s profit and job
creation? Will legalization increase migration pressures? Will deportation and border
control decrease legal immigration?
        To address these important questions we propose a new model representing two con-
nected labor markets, parameterized to match US and Mexico, and two groups of workers,
high-skilled and low-skilled, that are complementary in production. Firms create jobs
that are skill-specific and search frictions in the market exist. Legal and illegal migration
opportunities from Mexico to the US arise and people take them if they increase their
expected labor income net of costs. To focus on the issue of illegal immigrants we consider
migration of low-skilled workers only from Mexico, while US workers can be low-skilled
     Throughout the paper we will use the adjectives “legal” and “illegal” immigrants to characterize
immigrants who are endowed or not of proper documentation to reside and work in the US. Some scholars
refer to those groups as “regular” and “irregular” or as “documented” and “undocumented” immigrants.

(competing with immigrants) or high-skilled (complementing them in production). This
model incorporates aspects of labor markets and migration, which would not be captured
by a classical demand-supply framework and turn out to be crucial. First, we charac-
terize legal and illegal immigrants and native workers in the receiving country (US) as
potentially different in their outside options and in their probability of breaking up a
job-match. These differences affect the wage that each type of worker can bargain with
firms, for given productivity of the worker-firm match. In particular, illegal immigrants
usually have the worse outside option, followed by legal immigrants and then natives.
Hence the first group will accept lower wages, relative to legal immigrants and natives,
which imply that US firms can cut labor costs by hiring them. Second, as a consequence
of these labor cost savings, firms are willing to post more job openings and if those are
specific to skills, but not to immigrant workers, a positive job-creation effect will benefit
native employment opportunities too.
   Given the large productivity difference between Mexico and the US, illegal immigration
opportunities, albeit associated to worse conditions than legal ones, can be attractive to
Mexican unskilled workers. At the same time US firms benefit from illegal immigrants by
paying lower labor cost. These features capture the economic incentives that have lead to
illegal immigration in the US. However, there is also another crucial implication of this
framework: besides rich country skilled workers, also unskilled workers, can benefit from
illegal immigrants. More illegal workers push firms to create more jobs per unemployed
worker in the unskilled labor markets because their presence reduces the average firm’s
cost. As long as labor markets are not fully segmented between immigrants and natives,
natives will increase their employment too. Hence policies reducing the number of illegal
immigrants may cost employment and income to natives. With our model we can quantify
these costs and analyze how policies differ from each other.
   The four policies that we analyze are the following: (i) increasing border enforcement
to reduce illegal immigration opportunities (ii) increasing the costs that illegal immigrants
face in looking for a job (no access to benefits), (iii) increasing the frequency of depor-
tations and (iv) increasing the probability of legalization. In analyzing these policies we
take a status-quo driven approach. Rather than asking whether there is a theoretically
optimal number of illegal immigrants from the perspective of native income per person,
we consider the status quo, and we ask for each policy what would be the cost, in terms
of native income per person, wage and employment of reducing the number of illegal

immigrants by a certain percentage.
   The policies described can be separated into two categories. Three of them (increased
deportation, increased border control, and increased cost of looking for a job) not only
decrease the number of illegal immigrants, but they also reduce the total number of
immigrants (legal plus illegal). We will call these three “restrictive policies”. To the
contrary legalization, the fourth policy, decreases the number of illegal immigrants, but it
increases the number of total immigrants. By turning illegal immigrants into legal, this
policy leaves the total immigrant stock unchanged and it also provides stronger incentives
for potential immigrants, as more of them can become legal in the US.
   The three restrictive policies, by reducing unskilled immigrants (illegal and total)
have a depressing effect on wage and employment of skilled workers (complementary to
unskilled), and on firms’ profits (that benefit from the cost-reducing effects of illegal
immigrants). In the canonical model, however, they would increase employment and
wages of native unskilled, by reducing competition from unskilled immigrants. To the
contrary, because of the unskilled-job creation effect of immigrants described above, the
restrictive policies worsen labor market conditions for unskilled natives when analyzed
within our model. Legalization, instead, as it increases the total number of unskilled
immigrants enhancing their job-creation effect, produces a positive effect on wages and
employment of skilled natives and a positive effect also on unskilled native employment.
While the wage effect of legalization on unskilled natives is negative, the overall effect on
income per native in the receiving country is positive, contrarily to the restrictive policies
that reduce income per native in the receiving economy.
   Quantitatively our simulations shows the following effects. Increasing the deportation
rate of illegal immigrants or reducing illegal immigration opportunities at the border, to
achieve a 50% reduction in the number of illegal immigrants (a very aggressive program)
would produce an increase in the unemployment rate of unskilled natives by about 1.13%
of its initial value. The unemployment rate of native skilled workers also increases by
0.57%. The first effect is due to a decrease in unskilled job creation by firms and the
second to the negative productivity effect on skilled workers due to complementarity. A
similar result obtained through increasing the cost of unemployment for illegal immigrants,
would generate qualitatively similar but somewhat smaller results on unskilled native
unemployment (+0.95%). This is because that policy would also reduce the wage of
illegal immigrants and hence partly offset the negative incentives in creating unskilled

jobs. The same reduction in illegal immigrants achieved with a legalization program
would produce very different effects. The unemployment rate of unskilled natives would
decrease by about 1.31% of its initial value and that of skilled native would decrease by
1.20% of its initial value. The increase in legal immigrants generated by the legalization
program turns the negative labor market effect into a positive one. At the same time
legalization is the only policy that increases income per native in the scenario presented
above (+0.45%), while the losses of high skilled native workers, of firm profit and the
employment loss of low-skilled natives result in a net decrease in income per native when
adopting the other three policies (−0.25/ − 0.28%).
   Several checks on parameter values and on different scenarios about immigrants’ pro-
ductivity confirm that the above results are quite stable and they apply, qualitatively, to
most plausible scenarios. In summary, while the effects on income and unemployment
are quite small, the difference between the restrictive policies (that deliver similar effects
within each other) and the legalization is very clear: legalization is the only policy that
produces an increase in income per native and a decrease in native skilled and unskilled
unemployment. As the administrative costs to implement legalization are also likely much
smaller than those of increasing border security and certainly of those of deporting immi-
grants our analysis suggests that in terms of consequences on income of natives legalization
seems the best option.
   This paper is related to a large empirical literature on the effect of immigration on
US labor market outcomes (see the meta-analysis by Longhi, Nyikamp and Poot (2005),
(2008) and Lewis and Peri (forthcoming) for reviews of several important recent findings).
Most of that literature adopts a canonical neoclassical labor demand-supply approach to
derive a reduced form equation (e.g. Borjas 2003) or a slightly more structural approach
to estimate the elasticity of relative demand (Ottaviano and Peri 2012, Manacorda et al
2012). Very few studies analyze immigration within the context of search-matching models
of the labor market. Even fewer differentiate between legal and illegal immigration when
looking at labor market implications.
   The paper most closely related to ours is Chassamboulli and Palivos (2014). In that
paper, however, immigration is exogenous, only the receiving country is analyzed, only
legal immigrants exists and no policy is explicitly considered. Chassamboulli and Palivos
(2014) is the first paper, to our knowledge, that introduces the important job-creation
effect of immigrants stemming from the fact that the profit for the firm generated by

immigrants is larger than that generated by natives. This is an important building block
of our model too. We add to that framework the very important difference between
legal and illegal immigrants, the modelling of migration decision from Mexico, and the
representation and analysis of specific policies. With those tools we are the first to analyze
the income and employment impact of different policies reducing the number of illegal
   Palivos (2009) is one of the very few papers analyzing the welfare effects of illegal
immigrants on natives. Liu (2010) is the only other model we are aware of, that analyzes
the effects of illegal immigration on the receiving country using a search and matching
model. In his model Liu (2010) only includes illegal immigrants and assumes that they
are identical to natives in their search and labor supply behavior, but may be complemen-
tary to native workers in production. We consider, instead that immigrants, particularly
illegal ones, are disadvantaged relative to natives in terms of job search conditions and
search costs (they receive lower or no benefits when unemployed) and we also include the
possibility that illegal immigrants are subject to the risk of deportation. In our model
what is commonly referred to as “exploitation” of illegal immigrants, namely them be-
ing paid lower salaries, is due to their worse bargaining position vis-a-vis their employer
relative to natives.
   Finally, somewhat related to this paper, although mainly empirical, is the literature
on immigration and labor market institutions. It has been recognized for some time that
the specific labor market institutions (level of unemployment benefits, costs of hiring,
centralization of wage bargaining) can affect significantly the impact of immigration on
employment and wages of natives. For instance Angrist and Kugler (2003) show that
more protective labor markets result in larger impact of immigration on unemployment.
D’Amuri and Peri (2014) also show that labor reallocation and the complementarity effects
of immigrants can be larger in markets with lower rigidities.
   The rest of the paper is organized as follows. Section 2 presents the model and provides
intuition for its main results and the working of different mechanisms. We then describe
in Section 3 the policy experiments that we will be considering and two special cases
that allow us to illustrate the functioning of two important mechanisms in the model.
Section 4 describes the parameterization of the model calibrated to match the main labor
market statistics of the US and Mexico for the period between 2000 and 2010. Section 5
shows the main effects obtained by simulating four different policies that would achieve a

reduction of the number of illegal immigrants in the rich country. In Section 6 we present
some checks that the results are robust to reasonable variations of the parameter values.
Section 7 concludes the paper.

2     The Model
We describe here the main features of the model. Details of the equilibrium conditions
and derivation of intermediate results are described in the Appendix (A). We consider
two countries indexed by i = [1, 2]. Each country is endowed with a continuum of workers.
All agents are risk neutral and discount the future at a common rate r > 0, equal to the
real interest rate, time is continuous. Country 1 has higher wages and higher employment
rate than country 2. Hence workers have economic incentives to migrate from country 2
to country 1. No worker has incentives to migrate from country 1 to country 2. Migration
can be legal or illegal. We denote with I and L, respectively, the number of illegal and
legal migrant workers in country 1. The difference between the two is that opportunities
to migrate illegally are more frequent than those to migrate legally. However illegal
immigrants have higher search costs in the labor market and they face risk of deportation.
The size of the labor force native of country 1 (indicated as N ) is normalized to 1 and it is
divided into two types of workers: skilled in measure of S and unskilled in measure of 1−S.
Individuals born in country 2 are, instead, of measure F (foreign) and we assume that they
are all unskilled. The reason for this simplification is that we are focussing on the Mexico-
US migration which mainly involves unskilled workers (without tertiary education). The
total labor force of country 1 consists of natives, legal and illegal immigrants and its size
is 1 + I + L. The size of total labor force in country 2 is F − I − L. Individuals from
either country enter the labor force at rate τ and they exit at rate τ , so that the overall
size of the labor force (native of country 1 and 2) remains constant. The new individuals
enter the labor force as unemployed.
    At any point in time, opportunities to migrate arise as “random events” occurring
at rate µx , if the worker is unemployed in country 2, and at rate µex , if the worker is
employed. The subscript x = [I, L] indicates the type of the immigration opportunity.
Specifically, the worker may find an opportunity to migrate to country 1 legally (L) or
illegally (I). Once in country 1, illegal immigrants face some risk of deportation but they
may obtain legal status with probability n. This reflects the possibility that through
some special circumstances (e.g. marriage) some illegal immigrants may become legal.

This probability is however very small in absence of a legalization program. We assume
that µx > µex and without loss of generality we choose µex = 0, x = [L, I]. Migration
opportunities, that is, arise only for the unemployed, who are actively looking for them.
This captures the idea that, in order to migrate, workers often need to move closer to
the border and actively look for migration opportunities. A worker will act upon an
opportunity to migrate to country 1 if the benefit exceeds the cost. The migration cost,
z is heterogeneous across individuals and it is distributed according to the CDF Φ(z)
with support [z, z̄]. Only the fraction of workers with costs lower than expected benefits
will migrate. Once in country 1, migrants search for a job. Hence, the benefit from
immigrating to country 1 is the difference between the value of searching for a job as an
immigrant in country 1 and the value of searching for a job as a native in country 2.

2.1       Workers and Firms
Firms in country 1 operate in one of two intermediate sectors or in the final sector.2
The two intermediate sectors produce intermediate goods Y1u and Y1s using “unskilled”
and “skilled” labor, respectively. Each of these two sectors operates a linear technology,
which, through normalization of units, yields output equal to the number of the respective
workers employed. These intermediate inputs are non-storable. Once produced, they are
sold in competitive markets and are assembled for the production of country’s 1 final good
(Y1 ) which is also the numeraire. The production technology for the final good of country
1 is as follows:

                          Y1 = [α(Y1s )ρ + (1 − α)(Y1u )ρ ]1/ρ ,   ρ ≤ 1,               (1)

where α is a positive parameter that governs income shares and ρ determines the elasticity
of substitution between the unskilled and skilled inputs. Since the two intermediate inputs
are sold in competitive markets, their prices, ps1 and pu1 will be equal to their marginal
products, that is:                                 1−ρ
                                           =α            ,                              (2)
                                      u             Y1
                                     p1 = (1 − α)          ,                            (3)
      Our production side borrows from Acemoglu (2001).

The production technology in (1) implies diminishing marginal products and Edgeworth
complementarity between the two inputs Y1s and Y1u .3 The migrants from country 2 in
country 1 supply labor to the unskilled intermediate sector. The natives, on the other
hand, can be either skilled (s) or unskilled (u). Hence the skilled labor market in Country 1
hires only skilled native workers whose marginal productivity is ps1 and the unskilled labor
market hires unskilled native workers and immigrants with marginal productivity pu1 . The
production technology in (1) implies that immigrants are complements for skilled native
workers and perfect substitutes for unskilled native workers. Without loss of generality,
we keep the economy of country 2 simple by assuming that all workers in country 2 are
identically unskilled. There is therefore only one labor market in country 2 in which
all matches produce a constant output p2 and total output in that country is equal to
Y2 = (F − I − L − U2 )p2 , where U2 denotes the unemployed labor force of country 2 and
is defined below.

2.2       Search and Matching
In each labor market of country i unemployed workers and unfilled vacancies are brought
together via a stochastic matching technology Mi (Uit , Vit ), where t = [u, s] denotes the
skill-type. Uit and Vit denote, respectively, the number of unemployed workers and vacan-
cies of skill t in country i.4 We assume that the function Mi (Uit , Vit ), i = [1, 2] exhibits
standard properties: it is at least twice continuously differentiable, increasing in its argu-
ments, it exhibits constant returns to scale and satisfies the Inada conditions. Using the
property of constant returns to scale, we can write the flow rate of match per unemployed
worker of skill type t in country 1 as Mi (Uit , Vit )/Uit = mi (θit ). The flow rate of match
per vacancy is Mi (Uit , Vit )/Vit = qi (θit ), where θit = Vit /Uit = mi (θit )/qi (θit ) represents the
measure of tightness in market t of country i and mi (θit ) is increasing in θit while qi (θit ) is
decreasing in θit .
       Each firm posts at most one vacancy. The number of vacancies in each market is
determined endogenously by free entry. While vacancies in country 1 are skill-specific,
they cannot be specifically “targeted” to natives or to immigrants. They are open to
both native and immigrant workers with those skills. A vacant firm bears a recruitment
cost cti specific to the country and skill type, related to the expenses of keeping a vacancy
   3          ∂pt          ∂px
     That is: ∂Y1t < 0 and ∂Y1t > 0 for x 6= t.
                1             1
     Since there is only one labor market in country 2 the superscript t is not relevant in the case i = 2.
In what follows we therefore drop the superscript t whenever i = 2.

open and looking for a worker. An unemployed worker of type t in country i receives a
flow of income bti , which can be considered as the opportunity cost of employment. In
addition, unemployed workers pay a search cost πijt per unit of time where the subscript
j = [N, I, L] denotes the worker’s origin and status: native (N ), illegal immigrant (I) and
legal immigrant (L). Such subscript applies only to the unskilled market of country 1. We
account for the fact that a legal immigrant worker faces a higher search cost compared to
a native workers and an illegal immigrant faces even higher costs. The reason is that legal
immigrants, whether on temporary visas or permanent resident have access to significantly
fewer benefits than US citizens. Since the Personal Responsibility and Work Opportunity
Reconciliation Act (PRWORA) of 1996 many federal government benefits (Food stamps,
TANF, AFDC and others) were restricted to US citizens only. Hence non-naturalized legal
immigrants (the majority of unskilled foreign-born) had a significant larger cost of being
without a job. In the 2000’s some but not all, states re-instated some of them. Moreover
all legal immigrants on temporary visas (such as H2B and other working visas) are not
eligible for any welfare assistance, including unemployment insurance. Hence their access
to income when not employed is significantly smaller than for natives. Undocumented
immigrants cannot access any welfare program/unemployment insurance at all and hence
their cost of searching is even larger. We standardize the search cost of a native worker
                 s     u              u          u
to 0 and we set π1N = π1N = π2N = 0, π1I = πI , π1L = πL and we presume πI > πL > 0
which will be confirmed by the calibration.
   Legal immigrants face zero deportation risk. They have a positive probability of
returning home, however, reflecting the possibility of return for personal or other reasons.
Illegal immigrants face the additional risk of being repatriated by deportation. Hence the
return probability of illegal immigrants is higher than that of legal immigrants. Let dL
and dI denote the instant return rate of legal and illegal immigrants, respectively. We set
dI ≥ dL > 0 where their difference is the deportation rate. Upon return to country 2 the
worker joins the pool of unemployed and starts searching for a job.
   When a vacancy and a worker are matched, they bargain over the division of the
produced surplus. The status of the worker as well as the output that results from a
match are known to both parties. Wages, denoted as wij , differ by country (i), skill type
(t) and migration status (j). They are determined by Nash bargaining of the produced
surplus between the firm and the worker. After an agreement has been reached, production
commences immediately. Matches in country i dissolve at the rate σit , specific to skill

type t and country i. Following a job destruction, the worker and the vacancy enter the
corresponding market and search for new match.

2.3       Optimality Conditions and Free entry
At each point in time a worker is either employed (E) or unemployed (U ), while a vacancy
may be either filled (F ) or empty (V ). We use the notation Jijκ,t to denote the present
discounted value associated with each state κ = [V, F, U, E], where i = [1, 2] denotes the
country, j = [N, I, L] the worker’s immigration status and t = [u, s] indicates the worker’s
skill type.
       Eighteen Bellman equations describe the optimal behavior of workers and firms. Since
all workers and firms in country 2 are identical, four Bellman equations (one for each state
κ = [V, F, U, E]) describe the values of workers and firms in country 2. The remaining
fourteen Bellman equations describe the values of workers and firms in country 1, where
workers differ in terms of skills and immigration status. Specifically, for each of the three
states [F, U, E] there are four Bellman equations: one for legal immigrants, one for illegal
immigrants one for unskilled natives and one for skilled natives. The value of an unskilled
vacancy searching for a worker (V ), instead, is the same for legal immigrants, illegal
immigrants and unskilled natives because the vacancy is open to any of them and hence
it is described by the same Bellman equation. Another Bellman equation describes the
value of a skilled vacancy.5 The full set of Bellman equations is in the Appendix A.
       A second set of equilibrium conditions is that of free-entry (vacancy posting) on the
firm side in each of the two labor markets in country 1 (skilled and unskilled) and in
country 2. Firms open vacancies up to the point that an additional one has zero expected
value. In equilibrium this implies the following three conditions:

                           JiV,t = 0,    i = [1, 2] and t = [s, u] if i = 1,                          (4)

       Wages are then determined by Nash bargain between the matched firm and the worker.
The outside options of the firm and the worker are the value of a vacancy and the value
of being unemployed, respectively. Let Sijt ≡ JijF,t + JijE,t − (JijU,t + JiV,t ) denote the surplus
of a match between a vacancy of skill type t in country i and a worker of immigration
    The superscript t and the subscript j are not relevant for country 2, we therefore drop them whenever
i = 2. We also drop the superscript t in the cases j = [L, I], since all immigrants provide only unskilled
labor and can only be employed in unskilled jobs, and the subscript j in the case κ = V and i = 1, since
unskilled vacancies in country 1 are common to immigrants and natives.

status j. With Nash-bargaining the wage wij is set to a level such that the worker gets a
share β of the surplus, where β represents the relative bargaining power of workers, and
the share (1 − β) goes to the firm. This implies five equilibrium conditions (for matches
with legal immigrants, illegal immigrants, unskilled natives and skilled natives in country
1 and for matches with native workers in country 2) of the following form:

                  βSijt = JijE,t − JijU,t      (1 − β)Sijt = JijF,t − JiV,t              (5)
                  for i = [1, 2]; j = [N, I, L] if i = 1; and t = [s, u] if j = N

2.4    The Immigration Decision
An (unemployed) worker located in country 2 will choose to immigrate to country 1,
when an immigration opportunity arises, if its benefit exceeds its cost. The benefit from
migration is the difference between the value of searching for an unskilled job in country
1 and the value of searching in country 2. Workers are heterogeneous in their migration
costs. A worker whose migration cost is z, will chose to take advantage of an opportunity
to enter legally in country 1 only if J1L − J2U ≥ z while he/she will enter illegally if
J1I − J2U ≥ z. The threshold costs, denoted as zI∗ and zL∗ , and representing the highest
cost a worker is willing to pay in order to obtain illegal or legal entry into country 1, are
defined by the following conditions:

                                            zI∗ = J1I
                                                      − J2U                              (6)
                                            zL∗ = J1L
                                                      − J2U                              (7)

Notice that in equilibrium zL∗ > zI∗ because the value of searching for a job in country
                                                                           U     U
1 is higher when the immigrant is legal than when he/she is illegal (i.e. J1L > J1I ).
This proceeds from the assumptions that illegal immigrants have higher search costs
(πI > πL > 0) and face the risk of deportation (dI > dL > 0) both of which reduce the
value they can generate while searching for a job and the value of a job for them. This
implies that for a given distribution of the migration cost z, there will always be a larger
share of the country 2 population willing to take a legal immigration opportunity than
an illegal one.

2.5    The Steady-State conditions
The last set of equilibrium conditions are the steady-state conditions. Five of them
determine the constant number of unemployed workers of each type in each country by

equating the flows into and out of unemployment status for each type of worker: U2 are
               s                                     u
in country 2, U1N are skilled natives in country 1, U1N are unskilled natives in country 1,
U1L are legal immigrants in country 1 and U1I are illegal immigrants in country 1. Two
more conditions guarantee the stationarity of the number of legal and illegal immigrants,
L and I by equating the flows into and out of the group. The seven formal conditions
defining these steady state variables are given by (38-44) in the Appendix A.2. Let us
                               u     u
also define the variables φ ≡ U1N /(U1N + U1I + U1L ) to be the share of native workers in
the pool of unemployed unskilled workers of country 1 and λ ≡ U1L /(U1L + U1I ) to be the
share of legal immigrants among unemployed immigrants in country 1. In equilibrium φ
and λ are also constant. Writing the steady state conditions for unemployed and migrants
as a function of parameters, labor market tightness in the respective markets (θ1s , θ1u , θ2 )
and threshold costs zI∗ and zL∗ we obtain the following expressions:
                                         U1N      σs + τ
                                us1N =       = s 1                                         (8)
                                          S   σ1 + τ + m(θ1s )
                                        U1N      σu + τ
                               uu1N =       = u 1                                          (9)
                                        1−S  σ1 + τ + m(θ1u )
                                    U1I      σ u + τ + dI + n
                           u1I =        = u 1                                             (10)
                                     I   σ1 + τ + dI + n + m(θ1u )
                                 U1L   σ1u + τ + dL − n LI (1 − u1I )
                         u1L   =     =                                                    (11)
                                  L        σ1u + τ + dL + m(θ1u )
                                       U2         σ2 + τ
                             u2 =           =                                             (12)
                                    F −I −L   σ2 + τ + m(θ2 )

                                        µL Φ(zL∗ )u2 (F − I) + nI
                                L=                                                        (13)
                                         dL + τ + µL Φ(zL∗ )u2
                                           µI Φ(zI∗ )u2 (F − L)
                                 I=                                                       (14)
                                        dI + n + τ + µI Φ(zI∗ )u2
   Expressions (8)-(14) reveal some important mechanisms at work in our model. First,
(13) and (14) show that the equilibrium number of migrants I and L depend negatively
on the return probabilities (dI and dL ), positively on the rates of migration opportunities
(µI , µL ), and positively on the threshold migration costs zI∗ and zL∗ . The latter implies
that any economic and policy factor that increases the value of searching for a job in
country 1 relative to country 2, encourages immigration and translates in larger stocks of
legal L and illegal I immigrants in country 1. Second, the legalization rate (n) increases
the steady state number of legal immigrants L and decreases the steady state number of

illegal immigrants I. Third, as customary in these models, unemployment rates increase
with the relative separation probability σit and decrease with the matching probability
m(θit ) in the corresponding market.6 The impact of immigration policies on θit , and in
turn, on the matching probability m(θit ), is the main channel through which they can
influence the unemployment rate of the native workers that participate in that market.
                                                                         s     u
       Let us notice that once the constant equilibrium values of L, I, U1N , U1N , U1L , U1I are
determined then, a linear technology determines production of intermediates for country
1 so that: Y1u = 1 − S + L + I − U1N                            s
                                     − U1L − U1I and Y1s = S − U1N .

2.6       Equilibrium
The eighteen Bellman equations (20-37), five Nash-Bargaining conditions (5), three free
entry conditions (4), seven steady-state conditions (8-14) and two immigration-threshold
conditions (6-7) plus 2 marginal productivity conditions (2, 3), the two linear production
functions of intermediates and the aggregate production function of country 1 (1) and
country 2 constitute the fourty-one equilibrium conditions determining the fourty-one
endogenous variables of the model. These endogenous variables are the eighteen values
of Jijκ,t across countries, skills and immigration status, five wages (w1N
                                                                        s     u
                                                                           , w1N , w1L , w1I , w2 ),
three labor market tightness values (θ1u , θ1s , θ2 ) the number of unemployed and migrants
of each type (I, L, U1N    u
                        , U1N , U1L , U1I , U2 ) the immigration cost thresholds (zI∗ , zL∗ ) the
marginal productivity of skilled and unskilled workers (pu1 , ps1 ), the output of skilled and
unskilled firms (Y1u , Y1s ) and the final output of country 1 and 2 (Y1 , Y2 ). In the appendix
A.3 we show how to derive some intermediate results and provide a description for how
to solve the model in blocks. Given the fact, however, that some of the expressions are
cumbersome and not very intuitive we omit those from the text. We will explain, instead,
before calibrating and simulating the full model, the intuition behind two key mechanisms
with the help of two special cases described in Section 3 below.

2.7       Three key conditions
Before moving to the special cases, it is useful, to show three equilibrium relations that
provide some intuition for the role of legal and illegal immigrants on unskilled job creation
(vacancy posting) by firms in country 1.
     The unemployment rates of illegal and legal immigrants, u1I and u1L , increase also with the proba-
bility of return dI and dL (respectively) and with the exit/entry rate τ. All those parameters, in steady
state, act as separation rates.

Manipulating the Bellman equations of the value (to the firm) of a filled unskilled
          F,u    F       F
vacancy, J1N  , J1L and J1I we can write the difference in value between a native-filled
vacancy and a legal immigrant-filled one and between one filled by a legal and an illegal
immigrant as follows:

                                                 u         F,u
                         F,u    F      [w1L − w1N  ] + dL J1N
                        J1N  − J1L =                                                      (15)
                                          r + τ + σ1u + dL
                          F     F      [w1I − w1L ] + [dI − dL ] J1L
                         J1L − J1I   =                                                    (16)
                                           r + τ + σ1u + dI + n

Expression (15) reveals that if w1L < w1N , which would be the case when legal immigrants
                                                                  F,u    F
have higher search cost than natives (worse outside option) then J1N  < J1L as long as
dL is small. So the value of a legal immigrant is higher than that of a native to the firm
as long as, given their equal productivity, the wage paid to the immigrant is low enough
relative to the native wage, to compensate for the larger probability that the immigrants
ends the match because of return to the country of origin. Likewise condition (16) reveals
that if w1I < w1L , because illegal immigrants have worse outside options than legal ones,
      F     F
then J1L < J1I as long as the difference between the return probabilities (dI − dL ) which
represent the deportation rate, is sufficiently small. Hence, low deportation rates and high
search cost for illegal immigrants make them particularly valuable to the firm. And low
return rates and high search cost for legal immigrants make them valuable to the firm.
A negative value of expressions (15), (16) implies that legal and illegal immigrants may
stimulate job creation. This vacancy creation effect can be seen by manipulating the free
entry condition for unskilled vacancies in country one to get:

                         cu1         F,u
                                                    F               F
                                = φJ1N   + (1 − φ)  λJ 1L + (1 − λ)J1I                    (17)
                        q(θ1u )
   In this expression a larger share of immigrants among the unemployed (smaller value of
φ) and a larger share of illegal ones among them (smaller value of λ) increase the value of
the right-hand side, as long as (15) and (16) are negative, by shifting weight on J1I relative
to J1N  . This would imply more vacancy posting (free entry) and an increase in market
tightness θ1u to increase the left-hand side and reduce the right-hand side and maintain the
equality (recall that q(θ1u ) is decreasing in θ1u ). This implies that a policy that decreases
the share of both illegal and total immigrants in the labor force certainly depresses the
labor market tightness through this channel. However a policy that decreases the share of

illegal immigrants but increases the share of total immigrants may offset the first negative
impact with a positive impact on θ1u .
    Finally let us notice that the impact of immigrants on θ1u is also the channel through
which they affect skilled native workers. For as long as skilled and unskilled workers
are complementary in production, larger supply of the unskilled labor input Y1u , implies
larger price for the skilled labor input ps1 and thus larger profits for skilled firms. Hence
immigration policies that stimulate the creation of unskilled jobs and raise θ1u will also
stimulate the creation of skilled jobs (i.e. raise θ1s ), with a positive impact on skilled native
employment and wages.

3      Policy Effects in Special Cases
The rich structure of the model presented above allows us to analyze different policies. We
consider four of them: (i) reduced opportunities of illegal entry (increased border control)
captured by a decline in µI ; (ii) increased search cost for illegal immigrants, captured by
an increase in πI ; (iii) increased probability of deportation, captured by an increase of dI
for given dL (iv) increased probability of legalization, captured by an increase in n. All
these measures reduce the number of illegal immigrants. They have, however, different
implications on native labor markets as well as different incentive effects on immigration.
    There are the two main channels through which the presence of illegal (and legal)
immigrants affects labor market outcomes of natives in our model. The first channel,
that we call “price channel”, operates through the price of the intermediate input, pu1 .
As evidenced in equation (3) a decrease in I which is translated by the linear production
technology into a decrease in Y1u increases the marginal productivity of the unskilled labor
input thereby causing its price to rise. This “price effect” is the standard one, also present
in the canonical model: immigrants are substitute for native unskilled and reducing their
supply the marginal productivity of those increases putting upward pressure on their
wages and downward pressure on their unemployment rate. The second channel, that we
call “labor-cost channel” works instead through the expected labor cost to an unskilled-
sector firm from a filled job and follows the logic described in 2.7. A decrease in I which
corresponds to an increase in the share of legal immigrants λ would increase the expected
labor cost and reduce the value of a vacancy to an unskilled-sector firm. Hence firms post
fewer vacancies, the tightness of the labor market decreases putting downward pressure
on wages and upward pressure on unemployment of native unskilled.

For both effects it is important to know whether the policy reducing I also reduces
total immigrants (and their share in the labor force 1 − φ). A policy that decreases total
immigrants (I + L) together with I may exacerbate both effects, while a policy that
decreases I but increases (I + L) may attenuate and even reverse each effect. Before
considering the general case it is useful to consider two special cases in which the price
and the labor-cost effects work one at a time, while the other effect is muted.

3.1     Identical Options for Natives and Immigrants: the Price
        Channel only
The first case considered is one in which unskilled natives, legal and illegal immigrants
are identical in their search cost and in their probability of breaking up a match. The
parameter restrictions generating this case are: dI = dL = 0 (no probability of random
return for immigrants) and πI = πL = 0 (no search costs for immigrants). In this case a
decrease in I can be achieved through either border control or legalization (as the other
two channels have been muted) and it will essentially represent a decrease in the supply of
unskilled workers who are identical to native ones. While framed in a search-model with
two labor markets (skilled and unskilled) the working of this model is very similar to that
of a canonical model in which changing the number of illegal immigrants is like changing
the supply of unskilled workers. The effects on wages and employment are very similar
to what a classical model of labor demand and supply for two complementary types of
labor, would deliver.
   A consequence of the assumptions above is that legal immigrants, illegal immigrants
                                                  u     u     u
and native unskilled will be paid the same wage: w1N = w1L = w1I = w1t . Therefore,
the expected value of filling an unskilled vacancy with natives, legal or illegal immigrants
              F,u    F     F
is the same (J1N  = J1L = J1I ) and changing the share of legal, illegal immigrants and
natives in the labor force has no effect on the incentive to post vacancies (the right-hand
side of 17 does not depend on λ and φ in this case). This means that the labor-cost
channel is not operating and the only effects work through the price channel.

3.1.1   Identical Natives and Immigrants: Effects of Border Controls

A decrease in the number of illegal immigrants I achieved through increased border
control (lower µI ) reduces the total number of unskilled workers, (1 − S + L + I) in
country 1 and through the linear technology of the unskilled-sector it lowers Y1u =

m(θ1u )
σ1 +τ +m(θ1u )
 u               [1 − S + I + L] in equilibrium. Since skilled and unskilled labor inputs are
complements in the production of the final good (ρ < 1), the decrease in Y1u raises the
marginal productivity of unskilled labor pu1 and lowers that of skilled labor ps1 (from 2
and 3). Since higher prices lead to higher surplus of a match, this induces the posting
of unskilled jobs and raises the tightness and matching probability in the unskilled sec-
tor m(θ1u ). The increase in the matching probability of unskilled native workers, in turn,
drives their unemployment rate down and drives their wages up by improving their outside
option. The opposite holds for the skilled workers. Their unemployment rate increases
and their wage decreases.

3.1.2     Identical Natives and Immigrants: Effects of Legalization:

A decrease in the number of illegal immigrants I achieved through legalization (increase
in the rate n) leaves the total number of immigrants unchanged by simply increasing the
number of legal immigrants L by the same amount that it decreases illegal ones I. In
this case “legal”and “illegal” are simply labels given to identical type of workers and they
are also identical to unskilled natives. Hence legalization does not change any feature
of the labor market nor the incentives of people in country 2 to immigrate since there
is no benefit from obtaining legal status. Hence in this case the production and the
price of the unskilled intermediate input (Y1u and pu1 , respectively) remain unchanged. In
this case, the legalization of illegal immigrants has no impact on job creation and labor
market outcomes of native. Relative to the restrictive policy of increasing border controls,
legalization fully eliminates the positive effects on wage and employment of unskilled
natives and the negative effects on wage and employment of skilled native workers.

3.2      Perfect Substitution Skilled-Unskilled: the Labor-Cost chan-
         nel only
The second special case represents, in some respects, the opposite scenario. In this case
we consider perfect substitutability in production between skilled and unskilled workers
(which corresponds to the assumption ρ = 1 in the production function 1) but we maintain
differences between unskilled natives, legal immigrants and illegal immigrants so that
dI > dL > 0 and πI > πL > 0. Illegal immigrants can be deported and they have the
highest search costs. Legal immigrants have a certain probability of returning and also
intermediate search costs.

In this case, the price effect is muted because the prices (marginal productivity) of
the intermediate goods are constant, as the aggregate production function is linear in
the intermediates. In particular ps1 = α and pu1 = 1 − α and they will be unaffected
by the relative supply of skilled and unskilled. This implies that the skilled sector is
unaffected by the employment and labor market conditions in the unskilled sector, and as
a consequence, the wage and unemployment rate of skilled native workers are independent
of I. For unskilled workers, instead, the labor market effects of reducing illegal immigrants
works only through their effects on the expected labor cost. We can see from expression
(17) that an increase in the proportion of natives in total unemployment of country 1 (φ)
and an increase in the proportion of legal immigrants in the total number of unemployed
immigrants (λ) decreases the expected value of a vacancy and reduces job creation7 .
Moreover, policies that increase the search cost for illegal immigrants (πI ) or increase
their deportation probability (dI ) also influence directly the value of filling a vacancy
with an illegal immigrant J1I and in turn affect the right-hand side of (17). Policies
aimed at reducing illegal immigrants, therefore, can affect the expected labor cost to an
unskilled firm in country 1 and in turn their job creation.

3.2.1      Perfect Skill Substitution: Border controls, Search Cost and Deporta-
           tion Rates

Border controls, search cost and deportation rates reduce the total proportion of immi-
grants in the unemployment pool of country 1 hence increasing φ. This effect decreases
                      F            F                                          F
the weight on term [λJ1L + (1 − λ)J1I ] and increases the weight on the term J1N in the
                             F     F     F
right-hand side of (17). If J1I > J1L > J1N (which is the empirically relevant case) then
the decline in the proportion of immigrants will increase the expected labor cost to an
unskilled firm and decrease job creation and market tightness θ1u to maintain the equality
in (17).
      Also, immigration policies aimed at reducing illegal immigrants would, increase λ, the
fraction of legal workers among unemployed immigrants. Such a change shifts weight
      F      F                       F            F                            F     F
from J1I to J1L in the expression [λJ1L + (1 − λ)J1I ] of (17) and as long as J1I > J1L it
reduces market tightness θ1u to maintain equality. Both effects of the restrictive policies
conjure to a decrease in θ1u and hence they have an unambiguously positive impact on
unemployment and negative impact on wages of unskilled native workers.
      As long as expressions (15) and (16) are negative.

Besides this effect, increasing the search cost for illegal immigrants (πI ) or increasing
their deportation probability (dI ) affect directly the cost of employing an illegal immigrant
and hence his value to the firm, J1I . Those two policies have opposite effects on the cost
of employing an illegal immigrant. An increase in πI worsens the outside option of illegal
immigrants and hence lowers their wage w1I for given productivity thereby lowering labor
costs to the firm. The increased deportation policy, instead, by increasing the probability
of breaking a match, increases the cost of employing an illegal immigrant and hence it
reduces incentives for job posting. Hence, the same reduction in the number of illegal
immigrants achieved through an increase in πI , has a smaller negative impact on job
creation in the unskilled sector than an increase in dI , as it will increase J1I , while an
increase in dI will reduce J1I and have a further negative impact on unskilled labor market
tightness (via reducing the value of the right-hand side of 17).

3.2.2   Perfect Skill Substitution: Legalization

Following the three policies described above, both 1 − λ (the share of illegal immigrants
among unemployed immigrants) and 1 − φ (the immigrant share in unemployment) de-
crease and this produces a depressing effect on job creation. Legalization, instead, is the
only policy that may decrease the share of illegal immigrants without reducing the total
share of immigrants in country 1. By turning illegal into legal immigrants and increasing
the incentives to migrate, it actually increases the overall number of immigrants. The
positive effect on job creation implied by an increase in total immigrants will mitigate the
negative effect on job creation due to the reduction of illegal immigrants.
   The impact of legalization on the share of immigrants in the labor force (and in the
unemployment pool) is in general ambiguous. There are however reasonable parameter
configurations such that an increase in n raises total immigrants as share of the labor
force. This situation is more likely when the opportunities for legal entry µL are small.
To the limit when µL = 0 so that all new immigrants are illegal and can become legal
with probability n, an increase in the legalization probability raises the total number of
immigrants (legal and illegal together) for two reasons. First, because a higher legalization
probability means that the rate by which immigrants return home is on average lower
(fewer deportations). Second, because higher chances of legalization raise the expected
value of being illegal and this attracts a larger share of country 2 workers by increasing zI∗ .
In the general case, where µL > 0, a higher n will have the additional effect of deterring

the entry of legal immigrants through its negative impact on zL∗ .8 In our simulations for
the relevant parameter range an increase in n (legalization) lowers φ and raises λ. These
compositional changes involve two opposite effects on the creation of unskilled jobs in
country 1: the decrease in φ raises it, while the increase in λ lowers it. The relative size of
these two opposite effects depends on how large µL is relative to n among other factors.
Hence, while we cannot be sure that the effect of legalization through the labor-cost
channel is positive on the labor market tightness of unskilled, the described mechanisms
suggests that it will be larger (either reducing a negative effect or turning it into a positive
one) than the effect of the other policies considered.

3.3     Both Channels
In the basic model in which both channels are at work their relative effect will determine
the effect of reducing illegal immigrants on native unemployment and wages. The three
“restrictive” policies, reducing border crossing, increasing search costs and increasing
deportation have a positive price effect on employment and wages of native unskilled
workers, but they may also have a negative labor-cost effect on those variables. If the
second effect prevails they may be harmful to employment and wage of native unskilled.
They will certainly hurt the wage of skilled workers through the price channel. On the
other hand legalization, the only policy that may reduce the number of illegal immigrants
while increasing total immigration, attenuates the positive price effect on employment and
wages of unskilled but may have a positive impact through reduction of labor costs. In the
presence of a significant labor-cost effect, the effect of legalization on employment of native
unskilled may be beneficial, while it also benefits native skilled, if there is complementarity
across skills. Relative to the canonical model the search model introduces the important
labor-cost effects that may reverse or attenuate the canonical predictions on native labor
market outcomes. We will now simulate the effects obtained in a model matched to the
US-Mexican economy.
    A higher legalization probability improves the outside option of potential immigrants in country 2
and increases, in turn, their wage w2 . The value of searching for a job in country 2 (J2U ) therefore
increases, with a negative impact on zL  .

4         Baseline parameterization of the Model
We parameterize the model as to represent the average performance and conditions of
the US and the Mexican economy between 2000 and 2010 a period in which the presence
of illegal immigrants in the US peaked to about 11.5 million individuals. To do so we
combine three types of parameters. Some are taken from the literature. Others are taken
directly from the US and Mexican data. Finally a third group is chosen to match some
moments of the data. The parameter choice is summarized in Table 1. We describe here
in detail the sources and the methods used to calculate these parameters. For some key
parameters we perform robustness checks in Section 6 so as to test the sensitivity of our
main results to a range of plausible values.
        We use a Cobb-Douglas matching function, Mit = ξi (Uit )ε (Vit )1−ε , i = [1, 2], t = [u, s]
with constant return to scale to Uit and Vit . Following common practice in these models,
we set the unemployment elasticity of the matching function to ε = 0.5, which is within
the range of estimates reported in Petrongolo and Pissarides (2001). We postulate the
worker’s bargaining power to be β = 0.5, so that the Hosios condition (β = ε) is met
(see Hosios, 1990). We use the monthly interest rate r = 0.4% which implies a yearly
real rate of about 5%.9 This is calculated as the 30-year treasury constant maturity bond
rate minus the average GDP deflator over the period 1980-2010 for the US.10 We define
as skilled a worker who has at least some college education and unskilled workers are
those with no college education. Based on estimates from Ottaviano and Peri (2012)
the elasticity of substitution between workers with at least some college education and
workers with no college education is around 2. We therefore set ρ = 0.5. We assume
that the distribution of migration costs is uniform over the interval [0, z̄] where we have
standardized the lower bound to 0.
        We consider only immigrants from Mexico to the US whose vast majority is unskilled
(no college degree). Our measure of I + L therefore includes only unskilled Mexican
immigrants.11 The rest of the labor force of country 1 (which we normalize to 1) includes
unskilled and skilled US natives. The share of skilled workers in the US is set to S = 0.54.
This is the average (over years 2000 and 2010) share of US-born workers with some
     We match all the flow rates in the model to monthly rates.
     If one uses the short-term rate, namely the 3-months treasury rate during the 1980-2010 period, one
gets a smaller value of r = 0.2%. We use this in a robustness check. We consider the longer interval
1980-2010 as interest rates were unusually low in the 2000-2010 period.
     We omit the skilled Mexicans from our analysis with no consequences as they constitute a very small
percentage of the total.

college or more in the native working age (25-65) population. Data for this measure come
from IPUMS USA. Using the same data we find the monthly inflow of new individuals
in the US native labor force is 0.061% and hence we set τ = 0.00061. Using matched
data from the Current Population Survey (CPS) we estimated the average skilled and
unskilled monthly job-separation rates in the US (σ1s and σ1u , respectively) to be 0.024
and 0.032, respectively.12 As we are not aware of comparable estimates for Mexico, we set
the separation rate of Mexican jobs equal to that of unskilled US jobs, σ2 = σ1u = 0.032.
The Mexican population in working age (residing in Mexico and the US), F , is set to
0.33 of the US native population in working age which is standardized to 1. This number
equals the average value obtained from census data 2000 and 2010 by dividing the Mexican
unskilled population (in Mexico and US) and the total US native population in the US.
       From Masferrer and Roberts (2009), the total number of returnees to Mexico each
year (excluding deportation and averaged over the available period 2001-2005) was about
245,000 per year. These are the most precisely estimated returns to Mexico measured
during the 2000-2010 decade. As of 2001, the total Mexican-born unskilled population
in the US was about 9.1 millions.13 The basic yearly return migration rate for Mexican
migrants can be obtained as the ratio of returnees to US residents which equals 0.027 per
year. We consider this to be the “basic” rate of return for Mexican immigrants and we
apply it to legal Mexican immigrants for the decade 2000-2010. In order to compute the
yearly return rate of illegal Mexican immigrants we add to the basic rate the deportation
rate of non-criminal Mexicans. More specifically, applying the same basic return rate
of 0.027 to the illegal Mexican population in the US, which was estimated at about 5.2
million in 2001 (Passel and Capps, 2004), gives an estimate of 0.14 million of illegal
Mexicans returning to Mexico each year. We then add the deportation of non-criminal
Mexicans to that number by using Masferrer and Roberts (2009). They report, on average
(for the period 2001-2005), about 100,000 non-criminal Mexicans deported per year14 so
that the total number of previously illegal Mexicans going home (either returning or
deported) was about 0.24 million per year. The ratio of total returnees (0.24 million)
to the total number of illegal Mexicans (5.2 million) gives the return+deportation rate
of the illegal Mexicans equal to 0.0453 yearly. Based on these values and recalling that
     These measures include employment to unemployment and employment to inactivity transitions.
     This number comes from the US Census, 1990.
     As in our model people are deported while working or looking for a job we assume that they are not

You can also read